= Chebyshev polynomial
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The first-kind <Chebyshev polynomial> $T_n$ converts a multiple-angle <cosine> into a <polynomial>:
$$
T_n(\cos\theta)=\cos(n\theta),\qquad 0\leq\theta\leq\pi,\quad n\geq0.
$$
The <cosine addition formula> gives the <recurrence relation>
$$
T_0(x)=1,\qquad T_1(x)=x,\qquad T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)\quad(n\geq1).
$$
This constructs <polynomials> valid for every real $x$, including outside $[-1,1]$, where the real-angle description is unavailable. For example,
$$
T_2(x)=2x^2-1,\qquad T_3(x)=4x^3-3x,\qquad T_4(x)=8x^4-8x^2+1.
$$
For $n\geq1$, $T_n$ has <polynomial degree> $n$ and leading coefficient $2^{n-1}$. On $[-1,1]$ its <uniform norm> is one, even as its oscillations become more numerous.https://dlmf.nist.gov/18.5#i[T. H. Koornwinder and collaborators, NIST Digital Library of Mathematical Functions, §18.5(i), equation 18.5.1: trigonometric representation]{ref}https://dlmf.nist.gov/18.9#i[Koornwinder and collaborators, NIST DLMF, §18.9(i), Table 18.9.1: recurrence coefficients]{ref}
The <polynomial roots> of $T_n$ are especially simple:
$$
x_k=\cos\!\left(\frac{(2k-1)\pi}{2n}\right),\qquad k=1,\ldots,n.
$$
They correspond to equally spaced angles, but their locations in $x$ cluster near $-1$ and $1$. For $T_3$, the <polynomial roots> are $\sqrt3/2,0,-\sqrt3/2$. For $T_8$, the positive <polynomial roots> are approximately $0.980785,0.831470,0.555570,0.195090$, with their negatives completing the set. The distance from the right endpoint to the nearest <polynomial root> is $1-\cos(\pi/(2n))\sim\pi^2/(8n^2)$; central neighboring <polynomial roots> are separated on the larger scale $1/n$. The small change of <cosine> near angles zero and $\pi$ explains the clustering.
\Image[/gaussian-quadrature-chebyshev-root-projection.png]
{title=Analytical illustration of the eight roots of T8: equally spaced angles on a semicircle project to endpoint-clustered interpolation nodes}
These <polynomial roots> supply $n$ <interpolation nodes> for a <polynomial interpolation> of <polynomial degree> at most $n-1$. They differ from the $n+1$ <Chebyshev-Lobatto nodes> $\cos(j\pi/n)$, which include both endpoints. This choice of <polynomial roots> has nodal <polynomial>
$$
\prod_{k=1}^n(x-x_k)=2^{1-n}T_n(x).
$$
The <monic Chebyshev extremal polynomial> minimizes the <uniform norm> of this product among <monic polynomials> of <polynomial degree> $n$ on $[-1,1]$. Consequently, if $f$ has $n$ continuous <derivatives>, the usual <polynomial interpolation error> gives the useful bound
$$
\|f-p\|_\infty\leq\frac{2^{1-n}}{n!}\max_{-1\leq x\leq1}|f^{(n)}(x)|,\qquad\deg p\leq n-1.
$$
This optimizes the nodal factor in that bound, not the actual error for every <function>. Endpoint clustering helps control the large endpoint oscillations of <equally spaced interpolation> associated with <Runge's phenomenon>. The two node conventions and their geometric construction are discussed in Trefethen's authored text.https://people.maths.ox.ac.uk/trefethen/ATAP/ATAPfirst6chapters.pdf[Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Chapters 2–3 and Exercise 2.4: endpoint clustering, first-kind roots and polynomials]{ref}
The same <cosine substitution for polynomial approximation> explains the weighted <orthogonality>:
$$
\int_{-1}^1\frac{T_m(x)T_n(x)}{\sqrt{1-x^2}}\,dx=\int_0^\pi\cos(m\theta)\cos(n\theta)\,d\theta=\begin{cases}0,&m\ne n,\\\pi,&m=n=0,\\\pi/2,&m=n\geq1.\end{cases}
$$
The endpoint singularities of the weight are integrable. Thus the <Chebyshev polynomials> are <orthogonal polynomials> for this weight, making coefficients in a <least-squares polynomial in an orthogonal-polynomial basis> separate cleanly. Their <polynomial roots> also give <Chebyshev–Gauss quadrature>, connecting <polynomial interpolation> to <Gaussian quadrature>.<NIST Digital Library of Mathematical Functions/18 Orthogonal Polynomials>[NIST Digital Library of Mathematical Functions; Koornwinder and collaborators, NIST DLMF, §18.3, Table 18.3.1: first-kind Chebyshev weight and normalization]{ref}
Accuracy still depends on the target <function>. The <absolute value function> $|x|$ has a corner at zero, so the hypothesis requiring continuous <derivatives> in the displayed <polynomial interpolation error> bound fails. Endpoint clustering cannot remove that corner. A jump poses a stronger obstruction: <polynomials> are <continuous functions>, so they cannot converge in the <uniform norm> to a discontinuous <function> on the whole <interval>. Chebfun's documented examples show persistent overshoot for such jumps; these are a limitation of global <polynomial interpolation>, not a universal promise of rapid accuracy from <Chebyshev polynomials>.https://www.chebfun.org/docs/guide/guide04.html#4.3[Chebfun development team, Chebfun Guide, §4.3: polynomial interpolation of discontinuous functions]{ref}
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